Abstract
Integral inequalities that provide explicit bounds on unknown functions have proved to be useful in the study of qualitative properties of the solutions of differential, integral, and integro-differential equations.
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References
T.H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations. Ann. Math. (2) 20(4):292–296 (1919)
R. Bellman, The stability of solutions of linear differential equations. Duke Math. J. 10, 643–647 (1943)
A. Abdeldaim, A.A. El-Deeb, Some new retarded nonlinear integral inequalities with iterated integrals and their applications in retarded differential equations and integral equations. J. Fract. Calc. Appl. 5(suppl. 3S):Paper no. 9, 9 (2014)
A. Abdeldaim, A.A. El-Deeb, On generalized of certain retarded nonlinear integral inequalities and its applications in retarded integro-differential equations. Appl. Math. Comput. 256, 375–380 (2015)
A. Abdeldaim, A.A. El-Deeb, On some generalizations of certain retarded nonlinear integral inequalities with iterated integrals and an application in retarded differential equation. J. Egypt. Math. soci. 23, 470–475 (2015)
A. Abdeldaim, A.A. El-Deeb, On some new nonlinear retarded integral inequalities with iterated integrals and their applications in integro-differential equations. British J. Math. Comput. Science 5(4), 479–491 (2015)
D. Bainov, P. Simeonov, Integral Inequalities and Applications. (1992)
P. R. Beesack, Gronwall Inequalities. Carleton Mathematical Lecture Notes, (11) (1975)
A.A. El-Deeb, R.G. Ahmed, On some generalizations of certain nonlinear retarded integral inequalities for \(v\)olterra-\(f\)redholm integral equations and their applications in delay differential equations. J. Egypt. Math. Soc. 25, 279–285 (2017)
A.A. El-Deeb, R.G. Ahmed, On some explicit bounds on certain retarded nonlinear integral inequalities with applications. Adv. Inequalities Appl. p 19 (2016) (ISSN: 2050-7461)
H. Mostafa El-Owaidy, A.A. Ragab, A.A. El-Deeb, W. Mostafa Kamal Abuelela, Hassan Mostafa El-Owaidy, On some new nonlinear integral inequalities of Gronwall-Bellman type. Kyungpook Math. J. 54(4), 555–575 (2014)
B.G. Pachpatte, Integral and Finite Difference Inequalities and Applications (Elsevier B.V., 2006)
B.G. Baburao, Explicit bounds on certain integral inequalities. J. Math. Anal. Appl. 267(1), 48–61 (2002)
B.G. Pachpatte, A note on Gronwall-Bellman inequality. Math. Anal. Appl. 44, 758–762 (1973)
B.G. Pachpatte, Integral inequalities of Gronwall-Bellman type and their applications. Math. Physical Sci. 8, 309–318 (1974)
B.G. Pachpatte, On some integral inequalities and their applications to integrodifferential equations. Indian J. Pure Appl. Math. 8, 1157–1175 (1977)
B.G. Pachpatte, On some integral inequalities of the Gronwall-Bellman type. Indian J. Pure Appl. Math. 6, 769–772 (1975)
H. El-Owaidy, A. Abdeldaim, A.A. El-Deeb, On some new retarded nonlinear integral inequalities and their applications. Math. Sci. Lett. 3(3), 157–164 (2014)
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El-Deeb, A.A. (2018). A Variety of Nonlinear Retarded Integral Inequalities of Gronwall Type and Their Applications. In: Agarwal, P., Dragomir, S., Jleli, M., Samet, B. (eds) Advances in Mathematical Inequalities and Applications. Trends in Mathematics. Birkhäuser, Singapore. https://doi.org/10.1007/978-981-13-3013-1_8
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